## Tuesday, August 30, 2016

### ## Climate feedbacks and circuits

I've been arguing again at WUWT. Lord Monckton has been writing a series of articles (see also here, here and here), with more promised, on feedbacks and what he calls the "official equation". Mentioning feedbacks brings out all the engineers talking about feedback and instability. In the course of arguing, I think I see some general confusions arising from inadequate specifications of what circuitry is envisaged, and other general unsoundness. So I thought I could try to clear that up here, and in the process show an actual circuit which would implement the "official equation".

The first thing I try to emphasise here is that climate is not a circuit, and feedbacks are not used in GCM's. They are not the basis of climate science; in fact climate scientists talk about them far less than people imagine. Feedbacks are diagnostic tools - inferred from model output (or climate data) to help understanding. And you are free to imagine any kind of circuit or other apparatus that you think helps. That is a starting point - people are not always imagining the same thing, but they use the same vocabulary.

The basic concept here is climate sensitivity (CS). If you add a warming heat flux, usually from greenhouse effect, how much will the temperature rise? To make this more definite, it is often expressed as equilibrium CS (ECS). If you add a flux and then keep it constant, how much will temperature have risen by the time it has settled to steady state?

A starting point is what can be called Planck sensitivity. We know from the Stefan-Boltzmann law that a warmer planet will radiate more heat, and that will give a relation between flux and temperature change. For a black body, flux F=σT4 (S-B, with σ as the S-B constant). I propose to make F analogous to current and T to voltage, so this gives CS = dT/dF = 1/(4σT3) K/(W/m2) This gives it the units of resistance (analogy). With T=255 K, the effective radiating temperature of Earth, that would be 0.26. Earth is not a simple body - the atmosphere has an effect, and GCMs say that the right figure is about 0.31 (Soden and Held, 2006).

But ECS is generally reckoned to be a lot higher, because of the effects of positive feedbacks, especially water vapor feedback. Discussions of these are in S&H just cited, or Roe 2009. So then come the claims that positive feedback is necessarily unstable. It isn't, because it in effect adds to the negative Planck feedback. But if it outweighed that (and any other negative feedbacks) then it would be. That is the basis for talk of tipping points and thermal runaway.

Lord M's posts have started with a statement of the "official equation" which is from Roe 2009. The first version was this:

It is correctly stated there, but was misused, being applied to recent data far from equilibrium. I protested there, so it appeared in later forms (noted here) that could be used to justify the misuse. λ0 is the Planck feedback described above (0.31 K/(W/m2)). The equation can also be written
ΔTeq = ΔF/( 1/λ0 – Σci)
where the c's are feedback coefficients, with the convention that positive feedback c is positive. For consistency with λ0 I will reverse that sign convention and write c0=1/λ0:
ΔTeq = ΔF/( c0 + Σci)

This is easier to interpret. The c's are conductances (but negative for positive feedbacks), and the Planck c0 is added equally to the others. In fact, the equation is formally just Ohm's law, with combined conductance c = c0 + Σci

That makes it clear that the Planck c0 has the same status as the other feedbacks, and so is properly called a feedback. It also suggests a simpler interpretation. The "official equation" is just Ohm's law for a combined resistance made up of the feedbacks connected in parallel (where conductances add), and the stability criterion is simply that the resistance is positive. The catch, of course, is that negative c's, associated with positive feedback, are not an everyday circuit element, so some active feedback loop is required to implement them.

 Operational amplifiers These were much discussed at WUWT. An op-amp is a common circuit component, being a high-gain DC amplifier, depicted as on right (Wiki), though the power pins VS are often not shown. Idealised, it has the following properties Output V responds with infinite gain to the difference between V+ and V-. Inputs have infinite input impedance - ie no current can enter Output has zero impedance - any current can be withdrawn without affecting voltage.

One commenter at WUWT who wrote well about op-amps and feedback was Bernie Hutchins. He provided a good set of notes. Another commenter linked to these. I'm going to use one of Bernie's circuits; the basic element is an inverting amplifier:

This works as a see-saw; the input node is held at V=0, and the current entering the input must be Vin/Rin+Vout/Rf = 0 (input impedance infinite). If the R's are equal, Vout=-Vin. Bernie showed a circuit from his notes which uses two such stages to produce a non-inverting output, which can then be fed back as positive feedback:

You can see the experimental readings in light green - the circuit is clearly realisable and stable. The positive feedback increases the gain (ratio of Vout to Vin) from 1 to 3. Coincidentally, this is about the change effected by positive feedback in climate.

I promised a circuit which would implement the "official equation". I'm going to riff on Bernie's circuit. Vin and the initial R determine the entry current i, which corresponds to flux. I've let the first stage have general resistance R1 and the red feedback be R2, giving

The second stage operates as a simple voltage inverter no matter what the value of R. The reason is that the R's connect two outputs of zero impedance - ie they can supply any current without affecting voltage, and hence the rest of the circuit. So the relation between V and i is just got by summing currents at the input:
-V/R1+V/R2+i=0
or
V = i/(c1-c2) where c=1/R
And the loop gain is R1/R2 and is stable as long as that is less than 1, ie positive feedback c2<c1.

If c1 is now the sum of negative feedback conductances (connected in parallel) and c2 is the sum of positives, and ΔTeq~V, ΔF~i, we have the circuit emulating the "official equation" of equilibrium climate sensitivity with fedback.

So to clarify the issue of positive feedback and stability, it is true that an op-amp, say, with just positive feedback, of any amount, will be unstable. But what is usually meant by positive feedback in climate is feedback which is net positive when then added to the Planck feedback, giving negative overall. That addition increases the climate sensitivity (gain) without instability. But of course it makes it easier for a further perturbation to reach instability. Note that this special treatment of Planck feedback raises the question of whether it should then be regarded as a feedback at all. As I've shown, yes, it can with consistency. You can use terminology in which it is regarded as something else if you want.

Perhaps the simplest way of seeing stability is this. I've noted that the "official equation" has the form of Ohm's law, for a resistor. In math terms, it's isomorphic to an ohmic resistor. It must have the same properties. In particular, a resistor with positive resistance is quite stable. So then, with positive denominator, is the climate system described by the "official equation".

#### Equilibrium sensitivity

I spent a lot of time at WUWT trying to explain that ECS is an equilibrium analysis. The equation has no time information - it just says what happens when you get from one steady state to another. So it tells you nothing about states along the way, and can't be used to calculate them (as it was in Lord M's first post). Nor can you start drawing Bode plots (much demanded at WUWT) or whatever - that is frequency analysis and out of place in what is, in electrical terms, a DC (direct current) analysis. There is no frequency information to put on a Bode plot.

Update 1 Sept In the interests of relating to other ways in which people might express feedback, I should note two things.
1. I have formulated the circuit as current in - voltage out, because that corrsponds to climate sensitivity thinking. Feedback is more often discussed for a voltage-voltage, or maybe current-current, amplifier. The general way of thinking about this is via two=port networks. The simplification in this circuit is that both input and output have zero impedance (at input because the circuit must ensure that input voltage is zero for any current). So the only non-trivial ratio is z21, or y21 if you're thinking of the denominator.
2. I have used infinite gain amplifiers (op-amps). Feedback theory is often written in terms of finite gain A, as in:
V = Ai/(1 - A f)
You could regard the "official equation" as being that with A = λ0. So then it might be natural to regard the Planck term as the "pre-feedback gain" rather than a feedback. But then you can do the same algebra as above, dividing top and bottom by A, so 1/A becomes something added to the feedback terms, and I think this is more systematic. It reflects the fact that a finite gain amplifier and an op-amp with appropriate feedback are functionally equivalent. It really comes back to what you mean by "pre-feedback". And as I've said, these circuits are just analogues. You can choose.

1. Nice. Thank you (since no-one at WUWT will :-)

2. No one? I'm not sure.

I learned inbetween that an increasing amount of people there do at least think different than these noisy, arrogant and narrow-minded commenters you probably will have in mind. This 'minorité silencieuse' doesn't write much, but... reads a lot.

That's the reason why it would imho be great if Nick had time enough to write guest posts there about such topics he is tough in touch with.

1. FWIW, I agree. Amongst the closed minds are open ones

3. Nice.

As an EE, I've one minor point.

A physically realizable op amp with only positive feed back is a comparator with hysteresis. Yes, there are subtly different parts sold as comparators ( http://www.ti.com/lit/ds/symlink/lm311.pdf ), but real op amps can work as comparators.

An input above the threshold will produce an output at the positive limit. An input below the threshold will produce an output at the negative limit. This is stable, as long as there is positive feedback.

"It is a standard procedure to use hysteresis (positive feedback) around a comparator, to prevent oscillation, and to avoid excessive noise on the output because the comparator is a good amplifier for its own noise. "

http://www.ti.com/lit/an/snoa860/snoa860.pdf

1. Or a Schmitt trigger, which is a type of comparator. Its inventor, Otto Schmitt, was still an emeritus professor when I was going to the U of Minnesota. He was the lone occupant of the old Music Education building. The guy was nuts but many an engineering student would spend time listening to his stories and ideas. He would always carry two watches and was researching telekinesis at the end. http://www.ee.umn.edu/users/schmitt/bulletin.html

This feedback discussion is misguided. What you have to do is look at each case uniquely. The GHG warming via water vapor is a positive feedback system since as things heat up, more water vapor will go into the air. Yet the Arrhenius factor on outgassing has an interesting characteristic in that it won't spiral out of control. Set up the equations and you can estimate the setpoint that it will reach for a given initial forcing.

And of course, these don't have anything to do with classical electrical feedback circuits, except if you're an EE you have intuition on how the math works.

Some ideas are good and some ideas are bad.

2. Phil,
Thanks. Yes, I mentioned at WUWT that I spent a lot of time as a student messing with circuits to make music. That's probably why I related to Bernie Hutchins commentary - he's from the world of Moog, who was a hero of the times. Astable multivibrators, which did that switching between states, were our staple circuit.

3. Forget the electronics analogies, this is how you figure out sensitivity with the outgassing feedback that we know occurs:

http://theOilConunDrum.blogspot.com/2013/03/climate-sensitivity-and-33c-discrepancy.html

I am not going to reproduce this here, because the commenting markup is the crudest and ugliest that I have ever seen.

4. A "Schmitt trigger is a comparator circuit with hysteresis", according to Wikipedia. Might be more correct to say that a comparator circuit is a Schmitt trigger with or without hysteresis.

And yes, a warm smile in memory of Otto Schmitt. Thanks.

5. Oh yeah, for what it's worth ECS circuits would require a critically damped delay between input and output. Doable with op amps but hard to eliminate oscillations.

1. Oscillations! Likely all the oscillations that we see in the climate system are externally forced. The obvious ones are daily and seasonal, but then we have the behaviors such as QBO, which are likely forced by lunar gravitational cycles. The fact that consensus science views QBO as an emergent resonant behavior is troubling. Geoffrey Vallis is also questioning the complicated models we are using and that without a solid foundation of geophysical models "the edifice will come tumbling down".

Discussion here:
http://contextearth.com/2016/09/03/geophysical-fluid-dynamics-first-and-then-cfd/

6. I'd like to point out that Fig 6 shows absolutely no positive feedback. You have positive _GAIN_, not positive feedback. If you had any positive feedback the output would oscillate or "stick" to one of the rails.

A comparator is stable only in the sense that its output is constrained by the supply rails. An ideal comparator with an infinite supply voltage would output an infinite voltage, whereas an opamp with negative feedback and an infinite supply would output a finite voltage related to its input and the gain.

7. My bad on the last comment. There is positive feedback in that circuit. Build it on breadboard and see it oscillate.

1. Bernie Evans, who wrote about the circuit, did build it. It did not oscillate. The measured voltages are marked in green.

2. There are two inversions so the feedback from there is positive. It sums with the negative from the first stage. If the sum total is positive it will oscillate. It not it will be less negative. The analogy to climate is straight forward.

3. Jaff Patterson said:
"The analogy to climate is straight forward. "

Bwaha! The problem with these EEs is that they think that the climate can be reduced to circuit diagrams, not realizing that the underlying differential equations don't match those of electronics.

Jeff Patterson has probably never solved a Mathieu equation or a Storm-Liouville equation and probably couldn't because he thinks the solution to any problem is a Bode plot, ha ha.

4. The point Nick is making with this circuit analogy that like the climate system, the net feedback is always negative but that a small amount of positive feedback can reduce the amount of negative feedback and thus increase the gain. It's a point well made in language EE can relate to. Evidently you have some disagreement with the analogy so perhaps you should try stating your objection clearly instead of showing all who care to look how an asshat behaves.

5. Jeff Patterson said:
"... how an asshat behaves."

Why don't you crawl back to your Trump hole and leave this place for scientific discussion?

6. "Why don't you crawl back to your Trump hole and leave this place for scientific discussion?"
Maybe Jeff has said things elsewhere that you object to, but his comments here do not seem unscientific.

7. The point Nick is making with this circuit analogy that like the climate system, the net feedback is always negative but that a small amount of positive feedback can reduce the amount of negative feedback and thus increase the gain. It's a point well made in language EE can relate to. Evidently you have some disagreement with the analogy so perhaps you should try stating your objection clearly instead of showing all who care to look how an asshat behaves.

8. Brilliant Nick. The mistake many make is not appreciating the unit conversion issue which you point out. When the gains aren't unitless many are led astray.

1. For the record I have never tried to "disprove AWG". My view is that CO2 emissions contribute to global warming but that the climate sensitivity to such emissions is very near the low end of the accepted range of estimations. We are about a decade away from having enough high quality data to determine the question once and for all so that the wisest policy decision is to do nothing that would disrupt the global economy or deprive the developing world to the cheap energy required to raise their standard of living to that we enjoy.

Your other rude comments are equal inane. I've never attempted a Bode analysis of the climate. I have speculated in the past that some of the natural variation exhibited in the GMST records may be due to systemic response to some external event. That analysis was LaPlacian which are just differential equations in the frequency domain. Do you have trouble with differential equations?

2. For the record I have never tried to "disprove AWG". My view is that CO2 emissions contribute to global warming but that the climate sensitivity to such emissions is very near the low end of the accepted range of estimations. We are about a decade away from having enough high quality data to determine the question once and for all so that the wisest policy decision is to do nothing that would disrupt the global economy or deprive the developing world to the cheap energy required to raise their standard of living to that we enjoy.

Your other rude comments are equal inane. I've never attempted a Bode analysis of the climate. I have speculated in the past that some of the natural variation exhibited in the GMST records may be due to systemic response to some external event. That analysis was LaPlacian which are just differential equations in the frequency domain. Do you have trouble with differential equations?

3. This is why this guy belongs in the Trump camp of clowns:
"... or deprive the developing world to the cheap energy required to raise their standard of living to that we enjoy."

Evidently doesn't understand that fossil fuels are a finite & non-renewable resource and that we should be moving toward alternative energy schemes independent of future AGW effects.

Can't stand people like Jeff Patterson cause all they do is regurgitate the talking points that they absorb from slobs like Ailes. They aren't here to add any value, but to push their agenda.

9. Nice. But to simulate the climate you need to add a lot of capacitors - ocean, land, atmosphere, clouds - they should each be modeled as a resister in series with a capacitor. Then the input current must not be DC - day/night and seasons must be considered. Under those conditions the system becomes much more complex and positive feedback stability may be loss.

1. Robert C,
Well, I think that would be an analogue GCM 😒. The main problem being that we're struggling to get the DC values.

But I do think it could be useful to put in a capacitor for ocean, which would shift to the next time scale down. There is a definition of effective CS which basically does that. For EffCS you adjust for the supposed known flux into oceans, to try to get an ECS estimate before the long ocean stage is done.

2. When considering gyroscopes, solving the non-spinning equations gives you no clue what will happen when you pull the string. Same for a synchronous AC motor, using a DC power source won't provide useful work (just lots of magic smoke).

The fact that a DC circuit can be stable with positive feedback says absolutely nothing about what happens in the real world. Its not just that the energy from the Sun fluctuates, but any noise - a butterfly flapping its wings - *could* cause severe instability. That is why negative feedback rules in the great majority of *stable* systems.

3. "any noise - a butterfly flapping its wings - *could* cause severe instability."
That's what I'm trying to say by showing that combining feedbacks is like adding a set of conductances, positive and negative. Instability is where the total is no longer positive. The butterfly would have to offer a lot of conductance to tip the balance. But it also says that there is nothing special about having positive feedback in the mix. All that really matters is the after feedback gain.

4. regarding a butterfly flapping its wings:

Most people don't understand the distinction between initial conditions and boundary conditions. In fact boundary conditions can act much like a recurring set of initial conditions, where the boundary condition acts like a guide to synchronize a behavioral state over time. A good example of a boundary condition is the daily cycle and the seasonal cycle.

What causes the problem is that people immediately think a boundary condition can only have something to do with spatial dimensions, which is understandable since a boundary is typically considered a physical location.

A good example is the ocean tide, which is primarily set by boundary conditions. Tides will never go out of sync with the moon and the sun since they will forever nudge the response of the ocean to keep in sync with the defined orbit. However, they also behave following spatial boundary conditions, especially noticed in certain geophysical locations where the tides are most pronounced.

A huge instability such as a tsunami will have no impact on the tide, and it will readjust and keep going where it left off after the tsunami subsided.

Now think about how that applies to ENSO and QBO. Nothing about either of these two phenomena suggests that they are primarily driven by initial conditions. External forces are obviously maintaining a stationary behavior for each of them.

What is interesting is that there is a current flurry of media activity surrounding a measured perturbation of the QBO. There is speculation that this perturbation signals a change in the behavior of the QBO -- some say possibly due to AGW or a strong ENSO effect. Now, all we have to do is watch the QBO in the next few years and find out whether is synchronizes back to the behavior it has exhibited over the last 60 years.

Of course no one really understands QBO, since it is based on a half-backed theory of the crank Richard Lindzen, but say that that it is stimulated by lunar tidal forces, it should re-align with that cycle within the next few years. That will demonstrate a solid boundary condition.

A forcing caused by a growing CO2 imbalance is also a boundary condition, just that this is a boundary condition that is not stable over time. That's what makes it a bit more tricky.